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Bregman Hyperplane Trees for Fast Approximate Nearest Neighbor Search

Bilegsaikhan Naidan (Department of Computer and Information Science, Norwegian University of Science and Technology, Trondheim, Norway) and Magnus Lie Hetland (Department of Computer and Information Science, Norwegian University of Science and Technology, Trondheim, Norway)

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Abstract

This article presents a new approximate index structure, the Bregman hyperplane tree, for indexing the Bregman divergence, aiming to decrease the number of distance computations required at query processing time, by sacrificing some accuracy in the result. The experimental results on various high-dimensional data sets demonstrate that the proposed index structure performs comparably to the state-of-the-art Bregman ball tree in terms of search performance and result quality. Moreover, this method results in a speedup of well over an order of magnitude for index construction. The authors also apply their space partitioning principle to the Bregman ball tree and obtain a new index structure for exact nearest neighbor search that is faster to build and a slightly slower at query processing than the original.

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2. Preliminaries

Let be our database of vectors. Let be a strictly convex differentiable function. The Bregman divergence (Bregman, 1967) between based on is defined as:

(1)

Here, denotes the gradient of the function at and denotes the dot product between and . In other words, a Bregman divergence measures the distance between and its first-order Taylor expansion. The definition is illustrated in Figure 1.